Question

Establish each identity.

$\ln \left|sec\theta +\tan \theta \right|+\ln |sec\theta -\tan \theta |=0$

Short answer

The left side of the given equation has been simplified to bring the right side.

Step-by-step solution

Step 1. Given Information

The given equation is$\ln \left|sec\theta +\tan \theta \right|+\ln |sec\theta -\tan \theta |=0$

Step 2. Calculation 

Consider the left side of the given equation and simplify to bring the right side in order to establish the equation.

$$\begin{gathered} \ln \left|sec\theta +\tan \theta \right|+\ln |sec\theta -\tan \theta |=\ln \left|\right(sec\theta +\tan \theta \left)\right(sec\theta -\tan \theta \left)\right| \\ =\ln \left|\right(se{c}^2\theta -{\tan }^2\theta \left)\right| \\ =\ln \left|1\right| \\ =0 \end{gathered}$$

Hence, Proved